Understanding the inner workings of complex machine learning models is a long-standing problem, with recent research focusing primarily on local interpretability. To assess the role of individual input features in a global sense, we propose a new feature importance method, Shapley Additive Global importancE (SAGE), a model-agnostic measure of feature importance based on the predictive power associated with each feature. SAGE relates to prior work through the novel framework of additive importance measures, a perspective that unifies numerous other feature importance methods and shows that only SAGE properly accounts for complex feature interactions. We define SAGE using the Shapley value from cooperative game theory, which leads to numerous intuitive and desirable properties. Our experiments apply SAGE to eight datasets, including MNIST and breast cancer subtype classification, and demonstrate its advantages through quantitative and qualitative evaluations.

A primary goal of computer experiments is to reconstruct the function given by the computer code via scattered evaluations. Traditional isotropic Gaussian process models suffer from the curse of dimensionality, when the input dimension is high. Gaussian process models with additive correlation functions are scalable to dimensionality, but they are very restrictive as they only work for additive functions. In this work, we consider a projection pursuit model, in which the nonparametric part is driven by an additive Gaussian process regression. The dimension of the additive function is chosen to be higher than the original input dimension. We show that this dimension expansion can help approximate more complex functions. A gradient descent algorithm is proposed to maximize the likelihood function. Simulation studies show that the proposed method outperforms the traditional Gaussian process models.

We introduce a new paradigm, $\textit{measure synchronization}$, for synchronizing graphs with measure-valued edges. We formulate this problem as maximization of the cycle-consistency in the space of probability measures over relative rotations. In particular, we aim at estimating marginal distributions of absolute orientations by synchronizing the $\textit{conditional}$ ones, which are defined on the Riemannian manifold of quaternions. Such graph optimization on distributions-on-manifolds enables a natural treatment of multimodal hypotheses, ambiguities and uncertainties arising in many computer vision applications such as SLAM, SfM, and object pose estimation. We first formally define the problem as a generalization of the classical rotation graph synchronization, where in our case the vertices denote probability measures over rotations. We then measure the quality of the synchronization by using Sinkhorn divergences, which reduces to other popular metrics such as Wasserstein distance or the maximum mean discrepancy as limit cases. We propose a nonparametric Riemannian particle optimization approach to solve the problem. Even though the problem is non-convex, by drawing a connection to the recently proposed sparse optimization methods, we show that the proposed algorithm converges to the global optimum in a special case of the problem under certain conditions. Our qualitative and quantitative experiments show the validity of our approach and we bring in new perspectives to the study of synchronization.

We present a generative model of images that explicitly reasons over the set of objects they show. Our model learns a structured latent representation that separates objects from each other and from the background; unlike prior works, it explicitly represents the 2D position and depth of each object, as well as an embedding of its segmentation mask and appearance. The model can be trained from images alone in a purely unsupervised fashion without the need for object masks or depth information. Moreover, it always generates complete objects, even though a significant fraction of training images contain occlusions. Finally, we show that our model can infer decompositions of novel images into their constituent objects, including accurate prediction of depth ordering and segmentation of occluded parts.

It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a Gauss--Markov prior is used. In this paper the class of $\nu$ times differentiable linear time invariant Gauss--Markov priors is considered. A taxonomy of Gaussian estimators is established, with the maximum a posteriori estimate at the top of the hierarchy, which can be computed with the iterated extended Kalman smoother. The remaining three classes are termed explicit, semi-implicit, and implicit, which are in similarity with the classical notions corresponding to conditions on the vector field, under which the filter update produces a local maximum a posteriori estimate. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness $\nu+1$. Consequently, using methods from scattered data approximation and nonlinear analysis in Sobolev spaces, it is shown that the maximum a posteriori estimate converges to the true solution at a polynomial rate in the fill-distance (maximum step size) subject to mild conditions on the vector field. The methodology developed provides a novel and more natural approach to study the convergence of these estimators than classical methods of convergence analysis. The methods and theoretical results are demonstrated in numerical examples.

Real-world problems often involve the optimization of several objectives under multiple constraints. Furthermore, we may not have an expression for each objective or constraint; they may be expensive to evaluate; and the evaluations can be noisy. These functions are referred to as black-boxes. Bayesian optimization (BO) can efficiently solve the problems described. For this, BO iteratively fits a model to the observations of each black-box. The models are then used to choose where to evaluate the black-boxes next, with the goal of solving the optimization problem in a few iterations. In particular, they guide the search for the problem solution, and avoid evaluations in regions of little expected utility. A limitation, however, is that current BO methods for these problems choose a point at a time at which to evaluate the black-boxes. If the expensive evaluations can be carried out in parallel (as when a cluster of computers is available), this results in a waste of resources. Here, we introduce PPESMOC, Parallel Predictive Entropy Search for Multi-objective Optimization with Constraints, a BO strategy for solving the problems described. PPESMOC selects, at each iteration, a batch of input locations at which to evaluate the black-boxes, in parallel, to maximally reduce the entropy of the problem solution. To our knowledge, this is the first batch method for constrained multi-objective BO. We present empirical evidence in the form of synthetic, benchmark and real-world experiments that illustrate the effectiveness of PPESMOC.

We propose spectral methods for long-term forecasting of temporal signals stemming from linear and nonlinear quasi-periodic dynamical systems. For linear signals, we introduce an algorithm with similarities to the Fourier transform but which does not rely on periodicity assumptions, allowing for forecasting given potentially arbitrary sampling intervals. We then extend this algorithm to handle nonlinearities by leveraging Koopman theory. The resulting algorithm performs a spectral decomposition in a nonlinear, data-dependent basis. The optimization objective for both algorithms is highly non-convex. However, expressing the objective in the frequency domain allows us to compute global optima of the error surface in a scalable and efficient manner, partially by exploiting the computational properties of the Fast Fourier Transform. Because of their close relation to Bayesian Spectral Analysis, uncertainty quantification metrics are a natural byproduct of the spectral forecasting methods.

In this paper, we consider the problem of certifying the robustness of neural networks to perturbed and adversarial input data. Such certification is imperative for the application of neural networks in safety-critical decision-making and control systems. Certification techniques using convex optimization have been proposed, but they often suffer from relaxation errors that void the certificate. Our work exploits the structure of ReLU networks to improve relaxation errors through a novel partition-based certification procedure. The proposed method is proven to tighten existing linear programming relaxations, and asymptotically achieves zero relaxation error as the partition is made finer. We develop a finite partition that attains zero relaxation error and use the result to derive a tractable partitioning scheme that minimizes the worst-case relaxation error. Experiments using real data show that the partitioning procedure is able to issue robustness certificates in cases where prior methods fail. Consequently, partition-based certification procedures are found to provide an intuitive, effective, and theoretically justified method for tightening existing convex relaxation techniques.

Path planning plays a crucial role in robot action execution, since a path or a motion trajectory for a particular action has to be defined first before the action can be executed. Most of the current approaches are iterative methods where the trajectory is generated iteratively by predicting the next state based on the current state. Moreover, in case of multi-agent systems, paths are planned for each agent separately. In contrast to that, we propose a novel method by utilising fully convolutional neural network, which allows generation of complete paths, even for more than one agent, in one-shot, i.e., with a single prediction step. We demonstrate that our method is able to successfully generate optimal or close to optimal paths in more than 98\% of the cases for single path predictions. Moreover, we show that although the network has never been trained on multi-path planning it is also able to generate optimal or close to optimal paths in 85.7\% and 65.4\% of the cases when generating two and three paths, respectively.

Motivated by the emerging needs of decentralized learners with personalized learning objectives, we present an Assisted Learning framework where a service provider Bob assists a learner Alice with supervised learning tasks without transmitting Bob's private algorithm or data. Bob assists Alice either by building a predictive model using Alice's labels, or by improving Alice's private learning through iterative communications where only relevant statistics are transmitted. The proposed learning framework is naturally suitable for distributed, personalized, and privacy-aware scenarios. For example, it is shown in some scenarios that two suboptimal learners could achieve much better performance through Assisted Learning. Moreover, motivated by privacy concerns in Assisted Learning, we present a new notion of privacy to quantify the privacy leakage at learning level instead of data level. This new privacy, named imitation privacy, is particularly suitable for a market of statistical learners each holding private learning algorithms as well as data.